There is no good way to pronounce 'van Sluis' in English, since the 'ui' sound does not exist. There are two ways: one is like in the Spanish 'Ruiz' (slu-is), which we do not use, the other is like the word 'sluice' (slou-se), which is how we pronounce it in English. This is also the direct translation of the word. 'Sluis' is the Dutch word for 'sluice' or 'water lock', a device to transport ships from one water level to the other. It basically is a box, filled with water, with doors at either end. The door open at one end, the boat goes in, doors close. Now the water level in the sluice raises or lowers until it reaches the same level as the other side. Then the other doors open and the boat continues.

The Netherlands is full of sluices. Check out our animated explanation, the city of Terneuzen, or the ones of the Panama Canal. Here is a picture of the Gatun Locks in Panana:
, click on it to enlarge it.

Here is the Webster's dictionary entry:

Sluice \Sluice\, n. [OF. escluse, F. ['e]cluse, LL. exclusa, sclusa, from L. excludere, exclusum, to shut out: cf. D. sluis sluice, from the Old French. See {Exclude}.] 1. An artificial passage for water, fitted with a valve or gate, as in a mill stream, for stopping or regulating the flow; also, a water gate or flood gate.
2. Hence, an opening or channel through which anything flows; a source of supply.
Each sluice of affluent fortune opened soon. --Harte.
This home familiarity . . . opens the sluices of sensibility. --I. Taylor.
3. The stream flowing through a flood gate.
4. (Mining) A long box or trough through which water flows, -- used for washing auriferous earth.
Sluice gate, the sliding gate of a sluice.

The word 'van' is part of the last name, it is a Dutch prefix. 'Van' means 'of'. There are many prefixes in Holland: het (the), van der (of the), in (in), in het (in the), op (on), and many, many more combinations.
In our case, Peter's ancestors probably lived near a sluice, or operated on. That's were the last name came from.

Some trivia: When Peter grew up, boats passing through a sluice would pay the sluice master, who would collect the money in a wooden shoe connected to a line and rod...

Some more trivia: When looking up a name in a Dutch phonebook, look under 'Sluis, Peter van' (this opposed to the Belgium phonebook: 'Van Sluis, Peter').
And yes: the 'v' in 'van' is lowercase, then a space, and then 'Sluis' with uppercase 'S'...