For most applications I recommend the Rubycon ZLJ series capacitor of the same size.  They are a newer series and rated for much longer endurance than the Nichicon HN series.  Due to availability they are also priced lower.
Another option is the larger, 820uF 25V rated Panasonic FM capacitor which I have described and linked below. They can replace a 6.3V rated original but they are physically bulkier.

For PC motherboard applications, these Nichicon HN might be preferred as a more exact match to the originals found in many motherboards.  However even in those applications, the Rubycon ZLJ capacitors I linked above will also work well.

These capacitors are sold in sets of 5. To get 10 capacitors, you would buy 2 of these "items".
Click here to see all of the capacitors I have listed for sale.  I will combine shipping - the shipping shown is only charged once per entire order.
To shop for multiple items, please use "Add to Cart" to add each item you would like.  When you're done adding items, check your cart and you will see all items listed together with combined shipping.

820uF 6.3V Rubycon ZLJ - This is a newer capacitor series from Rubycon which has greater long term endurance than the Nichicon HN.  However, the Nichicon HN has a lower ESR spec and is intended more specifically for computer motherboards.
For most applications I suggest the Rubycon ZLJ, as it's ESR is plenty low enough for almost all purposes and it has a higher endurance rating.
Many Pentium4 and later motherboards originally came with extremely low ESR capacitors similar to the Nichicon HN.

820uF 25V Panasonic FM - Panasonic capacitor whose ESR is very close (actually slightly lower) to the Nichicon HN motherboard capacitors listed here.  They also have a much higher endurance.  The compromise is that they are physically much larger.  They have a larger diameter of 10mm with 5mm lead spacing and they are 25mm tall, so they could interfere with expansion cards.  The larger 10mm diameter can cause fitment problems on many PC motherboards.  However, if you can fit them physically then they will perform very well.  They are rated for a maximum of 25V and can safely replace a 6.3V rated original.

1000uF 10V NIC NRSG 8x20mm - 1000μF Low-ESR capacitors from Nippon Industries Co. with a narrow 8mm diameter.  Rated for maximum of 10V, they can safely replace a 6.3V rated original.

1000uF 10V NIC NRSG 10x16mm - Same as the above but with alternate dimensions of 10mm diameter and shorter 16mm height.

All Capacitor Listings - A wide range of capacitor values are available at this link and all are eligible for combined shipping. They are sorted into categories by their capacitance value (μF).  If you want something that isn't listed above, please check here.

I combine shipping.  Shipping is only charged once per entire order (please use "add to cart" to combine your items into a single order).

Shipping is the same flat amount regardless of how many items are in your order.  Additional items in your order will ride for free.  Please make sure to add all the items to your order first before paying. The easiest way to do this is by adding your items with the "add to cart" option. This way eBay will automatically combine the items into a single transaction with combined shipping when you pay.  If there is any problem with the total, let me know and I can fix the invoice.

For simplicity, I charge a fixed amount for this shipping method, rather than attempting to calculate it per-order.  This exceeds the postage price on smaller orders.  This is partly to account for the cost of packaging and also to justify the time required to prepare International packages.