How much is an 1877 quarter dollar worth?

This really depends on condition and mint mark. Since no mint mark was indicated, I will assume none. As such, an 1877 quarter that was really worn or damaged might fetch only a couple dollars on ebay. At today's silver market, ($19.10 per ounce for silver), this coin would be worth $3.45. Now, if the coin were in VF condition, it might fetch $20-30 or so on ebay and of course an AU coin would be much more. Those sale prices pale in comparison to what a price guide might suggest. for example, in the Feb 2014 issue of COINS magazine, the following values for an 1877 quarter are given

G 27.50

VG 31.50

F 33.50

VF 40

XF 62.50

AU 150

MS60 255

MS65 $1775.00

These are typical retail prices, in other Words what you might expect to pay for this coin, NOT what a dealer might offer.

If the coin is cleaned, all bets are off for higher grade coins.